An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent. Using ${e}^{\mathrm{iF}}$ as a kernel, an integral transform relates the wave functions of the corresponding quantum systems. The function F reduces in the classical limit (\ensuremath{\Elzxh}\ensuremath{\rightarrow}0) to the generating function of the classical transformation. Conversely, starting with the classical form, F can be calculated in a recurrent fashion, order by order in powers of \ensuremath{\Elzxh}. For the canonical transformation that relates a particle moving in an exponential (Liouville) potential to a free particle, the effective quantum generating function is identical to its classical counterpart. The generalization to quantum field theory is possible using the Schr\"odinger wave-functional formalism.An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent. Using ${e}^{\mathrm{iF}}$ as a kernel, an integral transform relates the wave functions of the corresponding quantum systems. The function F reduces in the classical limit (\ensuremath{\Elzxh}\ensuremath{\rightarrow}0) to the generating function of the classical transformation. Conversely, starting with the classical form, F can be calculated in a recurrent fashion, order by order in powers of \ensuremath{\Elzxh}. For the canonical transformation that relates a particle moving in an exponential (Liouville) potential to a free particle, the effective quantum generating function is identical to its classical counterpart. The generalization to quantum field theory is possible using the Schr\"odinger wave-functional formalism.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsFeb 15, 198737 citations

Canonical stochastic dynamical systems

Tetsuya Misawa, Kunio Yasue

A canonical stochastic dynamical system for time-symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations. A certain class of stochastic dynamical systems gives a Hamiltonian formalism of Nelson’s stochastic mechanics. In a manner analogous to classical mechanics, the notions of a stochastic Poisson bracket and canonical transformation are introduced to the stochastic dynamical systems. It is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation.A canonical stochastic dynamical system for time-symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations. A certain class of stochastic dynamical systems gives a Hamiltonian formalism of Nelson’s stochastic mechanics. In a manner analogous to classical mechanics, the notions of a stochastic Poisson bracket and canonical transformation are introduced to the stochastic dynamical systems. It is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation.

Journal of Mathematical PhysicsNov 1, 198713 citations

Canonical Transformations in Quantum Mechanics

A. Anderson

arXiv:hep-th/9305054Annals of PhysicsJun 1, 199490 citations

Canonical transformations in quantum mechanics

Maciej Błaszak, Ziemowit Domański

arXiv:1208.2835Annals of PhysicsJan 2, 201317 citations

Entropy of ideal quantum gases as a Lie function

B Bruhn

We investigate the Lie series representation of the canonical transformations in a complex phase space. It is shown that any canonical mapping in the complex domain can be labelled by two different functions. One of these functions corresponds to an observable in the sense of classical mechanics. The second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics. In particular we show that the entropy of ideal quantum gases generates a special canonical transformation and, moreover, the entropy itself can be represented as a Lie function formed by a special characteristic function.We investigate the Lie series representation of the canonical transformations in a complex phase space. It is shown that any canonical mapping in the complex domain can be labelled by two different functions. One of these functions corresponds to an observable in the sense of classical mechanics. The second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics. In particular we show that the entropy of ideal quantum gases generates a special canonical transformation and, moreover, the entropy itself can be represented as a Lie function formed by a special characteristic function.

Journal of Physics A Mathematical and GeneralDec 21, 19931 citation

An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent. Using

{e}^{\mathrm{iF}}

as a kernel, an integral transform relates the wave functions of the corresponding quantum systems. The function F reduces in the classical limit (\ensuremath{\Elzxh}\ensuremath{\rightarrow}0) to the generating function of the classical transformation. Conversely, starting with the classical form, F can be calculated in a recurrent fashion, order by order in powers of \ensuremath{\Elzxh}. For the canonical transformation that relates a particle moving in an exponential (Liouville) potential to a free particle, the effective quantum generating function is identical to its classical counterpart. The generalization to quantum field theory is possible using the Schr\"odinger wave-functional formalism.